The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious either a reasonable amount of concentration leads to new ideas or methods of obvious interest, or else one just leaves the problem alone. 'Perfect numbers' certainly never did any good, but then they never did any particular harm.
More Quotes from J. E. Littlewood:
I constantly meet people who are doubtful, generally without due reason, about their potential capacity as mathematicians. The first test is whether you got anything out of geometry. To have disliked or failed to get on with other mathematical subjects need mean nothing much drill and drudgery is unavoidable before they can get started, and bad teaching can make them unintelligible even to a born mathematician.J. E. Littlewood
In passing, I firmly believe that research should be offset by a certain amount of teaching, if only as a change from the agony of research. The trouble, however, I freely admit, is that in practice you get either no teaching, or else far too much.
J. E. Littlewood
It is possible for a mathematician to be 'too strong' for a given occasion. He forces through, where another might be driven to a different, and possible more fruitful, approach. (So a rock climber might force a dreadful crack, instead of finding a subtle and delicate route.)
J. E. Littlewood
I read in the proof sheets of Hardy on Ramanujan 'As someone said, each of the positive integers was one of his personal friends.' My reaction was, 'I wonder who said that I wish I had.' In the next proof-sheets I read (what now stands), 'It was Littlewood who said...'
J. E. Littlewood
We come finally, however, to the relation of the ideal theory to real world, or 'real' probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory 'If this is what you want, try it it is not my business to justify application of the system that can only be done by philosophizing I am a mathematician'. In practice he is apt to say 'try this if it works that will justify it'. But now he is not merely philosophizing he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
J. E. Littlewood
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